发表机构
South China Normal University（华南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对带初始真空的三维可压缩Navier-Stokes/Cahn-Hilliard系统，通过引入时间无关残差、采用半离散Galerkin格式，首次证明了其强解的局部存在性与唯一性。
AI 中文摘要
我们在三维欧氏空间中有界光滑区域内，建立了可压缩Navier-Stokes/Cahn-Hilliard系统强解的局部存在性与唯一性，允许初始密度为零。初始数据满足动量方程和相方程的适当正则性与相容性条件。主要困难源于动量方程和密度加权相方程的同时退化，这使得构造相容的正密度近似解和一致初始估计变得复杂。我们通过在化学关系中引入与时间无关的残差克服了这一困难，该残差精确保持相相容性、不改变初始相场，且当近似参数趋于零时在H¹空间中消失。对于每个正密度近似解，我们采用仅离散相变量、保留完整输运方程和动量方程的半离散Galerkin格式构造局部强解，随后推导与密度下界无关的先验估计，该估计给出共同寿命并允许过渡到真空极限。利用考虑密度与化学势耦合的修正差能证明唯一性。据我们所知，这是三维可压缩Navier-Stokes/Cahn-Hilliard系统带初始真空的首个局部适定性结果。
英文摘要
We establish local existence and uniqueness of strong solutions to the compressible Navier-Stokes/Cahn-Hilliard system in a bounded smooth domain in $\mathbb R^3$, allowing the initial density to vanish. The initial data satisfy suitable regularity and compatibility conditions for the momentum and phase equations. The main difficulty arises from the simultaneous degeneracy of the momentum equation and the density-weighted phase equations, which complicates the construction of compatible positive-density approximations and uniform initial estimates. We overcome this difficulty by introducing a time-independent residual into the chemical relation. The residual preserves the phase compatibility exactly, keeps the initial phase field unchanged, and vanishes in $H^1$ as the approximation parameter tends to zero. For each positive-density approximation, we construct a local strong solution using a semi-discrete Galerkin scheme that discretizes only the phase variables and retains the full transport and momentum equations. We then derive a priori estimates independent of the density lower bound, which yield a common lifespan and allow us to pass to the vacuum limit. Uniqueness is proved using a modified difference energy that accounts for the coupling between the density and the chemical potential. To our knowledge, this is the first local well-posedness result for the three-dimensional compressible Navier-Stokes/Cahn-Hilliard system with initial vacuum.
